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De Lellis, C; Royer-Carfagni, G (2001). Interaction of fractures in tensile bars with non-local spatial dependence. Journal of Elasticity, 65(1-3):1-31 (2002).

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Abstract

We propose to determine the displacement field u:ℐ⊂ℝ→ℝ of a 1-D bar extended in a hard device by minimizing a non-local energy functional of the type
Π[u]:=∫ ℐ Uu ' (x)+1 K∑ x i ∈J u [u](x i )ρ(x-x i )dx+∑ x i ∈J u ϕ([u](x i )),
where K is a material parameter, [u](x i ) denotes the jump of u at x i and J u ⊂ℐ is the set of all jump points. For appropriate choice of the bulk energy U(·), of the surface energy ϕ(·) and of the weight function ρ(·), we prove an existence theorem for minimizers in the space SBV(ℐ) of special bounded variation functions, and we qualitatively discuss their form by investigating the corresponding Euler-Lagrange equations. We show that, for sufficiently large values of bar elongation, minimizers of the energy are discontinuous and, most of all, the non-local term [u](x i )ρ(x-x i ) influences the relative position among the jump points, a finding that is of crucial importance to reproduce the experimental evidence.

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
DDC:510 Mathematics
Uncontrolled Keywords:nonlinear elasticity - variational model - fracture mechanics - non-local model - damage
Language:English
Date:2001
Deposited On:29 Nov 2010 16:27
Last Modified:23 Nov 2012 15:54
Publisher:Springer
ISSN:0374-3535
Publisher DOI:10.1023/A:1016143321232
Related URLs:http://www.ams.org/mathscinet-getitem?mr=1916961
Citations:Google Scholar™
Scopus®. Citation Count: 1

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